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Tensors of domain-related equivalent structures
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In this section, TENSOR provides the symmetry-adapted form of a list of material tensors for a particular space group, as well as the tensors of the domain-related equivalent structures corresponding to a particular choice of the group of the high-symmetry phase (parent group) and the transformation between the parent group and the space group (low-symmetry group, subgroup of the parent group). On the one hand, the parent group, the space group and the transformation matrix between them must be specified. On the other hand, a tensor must be defined by the user or selected from the lists of known equilibrium, optical and transport tensors, gathered from scientific literature. If a standard space group is defined and a known tensor is selected from the lists the program will obtain the required tensor from and internal database; otherwise, the tensor is calculated live. The working setting is defined by the rules explained here. Live calculation of tensors may take too much time and even exceed the time limit, giving an empty result, if high-rank tensors, a lot of symmetry elements and/or rare settings are introduced.
Further information can be found here
Information about the selected tensor
• 1 st rank Electric polarization vector Pi
• Polar tensor invariant under time-reversal symmetry operation
• Intrinsic symmetry symbol: V
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Information about the selected tensor
• 1 st rank Electrocaloric effect tensor pi
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔS=piEi
• Relates Electric field E with Entropy variation ΔS
• Intrinsic symmetry symbol: V
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Information about the selected tensor
• 1 st rank Heat of polarization tensor ti
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔS=tiΔPi
• Relates Polarization vector P variation with Entropy variation ΔS
• Intrinsic symmetry symbol: V
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Information about the selected tensor
• 1 st rank Piezoelectric polarization tensor under hydrostatic pressure dijj
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi=-dijjp
• Relates Hydrostatic pressure p with Polarization vector P.
• This tensor of rank 1 is obtained from the contraction of the last two indices of the piezoelectric tensor dijk.
• Intrinsic symmetry symbol: V
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Information about the selected tensor
• 1 st rank Pyroelectric tensor pi
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔPi=piΔT
• Relates Temperature variation ΔT with Polarization vector P variation
• Intrinsic symmetry symbol: V
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Information about the selected tensor
• 1 st rank Axial toroidal moment Ai
• Axial tensor invariant under time-reversal symmetry operation
• Intrinsic symmetry symbol: eV
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Information about the selected tensor
• 2 nd rank Dielectric impermeability tensor βij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ei=βijDj
• Relates Electric displacement field D with Electric field E
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• βij = βji
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Information about the selected tensor
• 2 nd rank Dielectric permittivity tensor εij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Di=εijEj
• Relates Electric field E with Electric displacement field D
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• εij = εji
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Information about the selected tensor
• 2 nd rank Dielectric susceptibility tensor χeij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi=χeijEj
• Relates Electric field E with Polarization vector P variation
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• χeij = χeji
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Information about the selected tensor
• 2 nd rank Heat of deformation tensor αij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔS=αijεij
• Relates Strain tensor εij with Entropy variation ΔS
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• αij = αji
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Information about the selected tensor
• 2 nd rank Magnetic permeability tensor μij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Bi=μmijHj
• Relates Magnetic field H with Magnetic field B
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• μij = μji
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Information about the selected tensor
• 2 nd rank Magnetic susceptibility tensor χmij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Mi=χmijHj
• Relates Magnetic field H with Magnetization M
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• χmij = χmji
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Information about the selected tensor
• 2 nd rank Piezocaloric effect tensor βij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔS=βijσij
• Relates Stress tensor σij with Entropy variation ΔS
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• βij = βji
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Information about the selected tensor
• 2 nd rank Strain by hydrostatic pressure sijkk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: εij=-sijkkp
• Relates Hydrostatic pressure p with Strain tensor εij.
• This tensor of rank 2 is obtained from the contraction of the last two indices of the elastic compliance tensor sijkl.
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• sij = sji
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Information about the selected tensor
• 2 nd rank Strain tensor εij
• Polar tensor invariant under time-reversal symmetry operation
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• εij = εji
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Information about the selected tensor
• 2 nd rank Thermal expansion tensor αij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: εij=αijΔT
• Relates Temperature variation ΔT with Strain tensor εij
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• αij = αji
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Information about the selected tensor
• 2 nd rank Thermoelasticity tensor βij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=-βijΔT
• Relates Temperature variation ΔT with Stress tensor σij
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• βij = βji
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Information about the selected tensor
• 2 nd rank Toroidic susceptibility tensor τij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ti=τijSj
• Relates Toroidal field S with Toroidal moment T
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• τij = τji
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Information about the selected tensor
• 2 nd rank Magnetotoroidic tensor ζij (direct effect)
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Mi=ζijSj
• Relates Toroidal field S with Magnetization M
• Intrinsic symmetry symbol: eV2
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Information about the selected tensor
• 2 nd rank Magnetotoroidic tensor ζTij (inverse effect)
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Ti=ζTijHj
• Relates Magnetic field H with Toroidal moment T
• Intrinsic symmetry symbol: eV2
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Information about the selected tensor
• 3 rd rank Acoustoelectricity tensor ρijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=ρijkJk
• Relates Alternating electric current density J with Stress tensor σij
• Intrinsic symmetry symbol: [V2]V
• Symmetrized indexes due to intrinsic symmetry:
• ρijk = ρjik
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Information about the selected tensor
• 3 rd rank Isothermal piezoelectric tensor eTijk (inverse effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=eTijkEk
• Relates Electric field E with Stress tensor σij
• Intrinsic symmetry symbol: [V2]V
• Symmetrized indexes due to intrinsic symmetry:
• eTijk = eTjik
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Information about the selected tensor
• 3 rd rank Piezoelectric tensor dTijk(inverse effect) dTijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: εij=dTijkEk
• Relates Electric fieldE with Strain tensorεij
• Intrinsic symmetry symbol: [V2]V
• Symmetrized indexes due to intrinsic symmetry:
• dTijk = dTjik
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Information about the selected tensor
• 3 rd rank Second order magnetoelectric tensor αTijk (inverse effect) αTijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi=αTijkHjHk
• Relates Magnetic field H with Polarization P
• Intrinsic symmetry symbol: V[V2]
• Symmetrized indexes due to intrinsic symmetry:
• αTijk = αTikj
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Information about the selected tensor
• 3 rd rank Isothermal piezoelectric tensor eijk (direct effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Di=eijkεjk
• Relates Strain tensor εij with Electric displacement field D
• Intrinsic symmetry symbol: V[V2]
• Symmetrized indexes due to intrinsic symmetry:
• eijk = eikj
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Information about the selected tensor
• 3 rd rank Piezoelectric tensor dijk(direct effect) dijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi=dijkσjk
• Relates Stress tensorσij with Polarization vectorP
• Intrinsic symmetry symbol: V[V2]
• Symmetrized indexes due to intrinsic symmetry:
• dijk = dikj
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Information about the selected tensor
• 4 th rank Elastic compliance tensor Sijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: εij=Sijklσkl
• Relates Stress tensor σij with Strain tensor εij
• Intrinsic symmetry symbol: [[V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• Sijkl = Sjikl
• Sijkl = Sijlk
• Sijkl = Sklij
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Information about the selected tensor
• 4 th rank Elastic stiffness tensor Cijkl Cijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=Cijklεkl
• Relates Strain tensor εij with Stress tensor σij
• Intrinsic symmetry symbol: [[V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• Cijkl = Cjikl
• Cijkl = Cijlk
• Cijkl = Cklij
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Information about the selected tensor
• 4 th rank Viscosity tensor ηijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=ηijkl∂εkl/∂t
• Relates Strain tensor rate ∂εij/∂t with Stress tensor σij
• Intrinsic symmetry symbol: [[V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• ηijkl = ηjikl
• ηijkl = ηijlk
• ηijkl = ηklij
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Information about the selected tensor
• 4 th rank Damage effect tensor Dijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=Dijklσkl
• Relates Stress tensor σij (before damage) with Effective stress tensor σij (after damage)
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• Dijkl = Djikl
• Dijkl = Dijlk
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Information about the selected tensor
• 4 th rank Electrostriction tensor γijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: εij=γijklEkEl
• Relates Electric field E and Electric field E with Strain tensor εij
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• γijkl = γjikl
• γijkl = γijlk
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Information about the selected tensor
• 4 th rank Magnetostriction tensor Nijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: εij=NijklMkMl
• Relates Magnetization M and Magnetization M with Strain tensor εij
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• Nijkl = Njikl
• Nijkl = Nijlk
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Information about the selected tensor
• 4 th rank Flexoelectric New μijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi=μijkl∇jεkl
• Relates Strain tensor gradient ∇iεjk with Polarization vector P.
• According to Eliseev and Morozovska,Phys.Rev.B 98,094108 (2018) the intrinsic symmetry of this tensor is not V2[V2] but V[V3].This allows to further reduce the number of independent coefficients.
• Intrinsic symmetry symbol: V[V3]
• Symmetrized indexes due to intrinsic symmetry:
• μijkl = μilkj
• μijkl = μikjl
• μijkl = μijlk
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Information about the selected tensor
• 4 th rank Elastothermoelectric power tensor Eijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔΣij=Eijklεkl
• Relates Strain tensor εij with Thermoelectric power tensor variation ΔΣij
• Intrinsic symmetry symbol: V2[V2]
• Symmetrized indexes due to intrinsic symmetry:
• Eijkl = Eijlk
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Information about the selected tensor
• 4 th rank Flexoelectric tensor μijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi=μijkl∇jεkl
• Relates Strain tensor gradient ∇iεjk with Polarization vector P.
• Intrinsic symmetry symbol: V2[V2]
• Symmetrized indexes due to intrinsic symmetry:
• μijkl = μijlk
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Information about the selected tensor
• 4 th rank Piezothermoelectric power tensor Πijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ΔΣij=Πijklσkl
• Relates Stress tensor σij with Thermoelectric power tensor variation ΔΣij
• Intrinsic symmetry symbol: V2[V2]
• Symmetrized indexes due to intrinsic symmetry:
• Πijkl = Πijlk
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Information about the selected tensor
• 5 th rank Acoustic activity tensor bijklm
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=bijklm∇mεkl
• Relates Strain tensor gradient ∇lεij with Stress tensor σij
• Intrinsic symmetry symbol: [[V2][V2]]V
• Symmetrized indexes due to intrinsic symmetry:
• bijklm = bjiklm
• bijklm = bijlkm
• bijklm = bklijm
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Information about the selected tensor
• 5 th rank Second-order piezoelectric tensor dijklm
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi=dijklmσjkσlm
• Relates Stress tensor σij and Stress tensor σij with Polarization vector P
• Intrinsic symmetry symbol: V[[V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• dijklm = dikjlm
• dijklm = dijkml
• dijklm = dilmjk
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Information about the selected tensor
• 6 th rank Third order elastic compliance tensor Sijklmn
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: εij=Sijklmnσklσmn
• Relates Stress tensor σij and Stress tensor σij with Strain tensor εij
• Intrinsic symmetry symbol: [[V2][V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• Sijklmn = Sjiklmn
• Sijklmn = Sijlkmn
• Sijklmn = Sijklnm
• Sijklmn = -Sklijmn
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Information about the selected tensor
• 6 th rank Third order elastic stiffness tensor Cijklmn
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=Cijklmnεklεmn
• Relates Strain tensor εij and Strain tensor εij with Stress tensor σij
• Intrinsic symmetry symbol: [[V2][V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• Cijklmn = Cjiklmn
• Cijklmn = Cijlkmn
• Cijklmn = Cijklnm
• Cijklmn = -Cklijmn
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Information about the selected tensor
• 8 th rank Damage tensor Rijklmnpq
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Cijkl=RijklmnpqCmnpq
• Relates Elastic stiffness tensor Cijkl (before damage) with Elastic stiffness tensor Cijkl (after damage)
• Intrinsic symmetry symbol: [[[V2][V2]][[V2][V2]]]
• Symmetrized indexes due to intrinsic symmetry:
• Rijklmnpq = Rjiklmnpq
• Rijklmnpq = Rijlkmnpq
• Rijklmnpq = Rijklnmpq
• Rijklmnpq = Rijklmnqp
• Rijklmnpq = Rklijmnpq
• Rijklmnpq = Rijklpqmn
• Rijklmnpq = Rmnpqijkl
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Information about the selected tensor
• 8 th rank Fourth order elastic compliance tensor Sijklmnpq
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: εij=Sijklmnpqσklσmnσpq
• Relates Stress tensor σij and Stress tensor σij and Stress tensor σij with Strain tensor εij
• Intrinsic symmetry symbol: [[V2][V2][V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• Sijklmnpq = Sjiklmnpq
• Sijklmnpq = Sijlkmnpq
• Sijklmnpq = Sijklnmpq
• Sijklmnpq = Sijklmnqp
• Sijklmnpq = Sijklpqmn = Sijmnklpq = Sijmnpqkl = Sijpqklmn = Sijpqmnkl = Sklijmnpq = Sklijpqmn = Sklmnijpq = Sklmnpqij = Sklpqijmn = Sklpqmnij = Smnijklpq = Smnijpqkl = Smnklijpq = Smnklpqij = Smnpqijkl = Smnpqklij = Spqijklmn = Spqijmnkl = Spqklijmn = Spqklmnij = Spqmnijkl = Spqmnklij
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Information about the selected tensor
• 8 th rank Fourth order elastic stiffness tensor Cijklmnpq
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: σij=Cijklmnpqεklεmnεpq
• Relates Strain tensor εij and Strain tensor εij and Strain tensor εij with Stress tensor σij
• Intrinsic symmetry symbol: [[V2][V2][V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• Cijklmnpq = Cjiklmnpq
• Cijklmnpq = Cijlkmnpq
• Cijklmnpq = Cijklnmpq
• Cijklmnpq = Cijklmnqp
• Cijklmnpq = Cijklpqmn = Cijmnklpq = Cijmnpqkl = Cijpqklmn = Cijpqmnkl = Cklijmnpq = Cklijpqmn = Cklmnijpq = Cklmnpqij = Cklpqijmn = Cklpqmnij = Cmnijklpq = Cmnijpqkl = Cmnklijpq = Cmnklpqij = Cmnpqijkl = Cmnpqklij = Cpqijklmn = Cpqijmnkl = Cpqklijmn = Cpqklmnij = Cpqmnijkl = Cpqmnklij
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Information about the selected tensor
• 2 nd rank Index ellipsoid βij
• Polar tensor invariant under time-reversal symmetry operation
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• βij = βji
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Information about the selected tensor
• 2 nd rank Second-order thermo-optical effect tensor Tij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=Tij(ΔT)2
• Relates Temperature variation ΔT and Temperature variation ΔT with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• Tij = Tji
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Information about the selected tensor
• 2 nd rank Thermo-optical effect tensor tij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=tijΔT
• Relates Temperature variation ΔT with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• tij = tji
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Information about the selected tensor
• 2 nd rank Verdet tensor (related to Faraday effect) Vij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=εijmVmkHk(εijm, with εijk Levi-Civita axial antisymmetric tensor)
• Relates magnetic field H with the antisymmetric part of the dielectric impermeability tensor variation Δβij
• Related with Faraday effect coefficients F: Fijk=εijmVmk, where εijm Levi-Civita axial antisymmtric tensor.
• Intrinsic symmetry symbol: V2
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Information about the selected tensor
• 2 nd rank Optical activity tensor gij
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: G=gijlilj
• Relates Direction cosines li and Direction cosines lj with Optical activity coefficient G
• Intrinsic symmetry symbol: e[V2]
• Symmetrized indexes due to intrinsic symmetry:
• gij = gji
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Information about the selected tensor
• 2 nd rank Thermogyration tensor gij
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δgij=gijT
• Relates Temperature T with Optical activity tensor variation Δgij
• Intrinsic symmetry symbol: e[V2]
• Symmetrized indexes due to intrinsic symmetry:
• gij = gji
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Information about the selected tensor
• 3 rd rank Natural optical activity γijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=i γijkkk
• Relates light wave vector with dielectric impermeability tensor variation (antisymmetric part).
• Connected with gyrotropic second-rank axial tensor glk=k0/2εijkgijl, with εijk Levi-Civita axial antisymmetric tensor and k0 modulus of light wave vector in vacuum. Gyration given by G=glkklkk/k02.
• Real in non-disipative media.
• Intrinsic symmetry symbol: {V2}V
• Symmetrized indexes due to intrinsic symmetry:
• γijk = -γjik
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Information about the selected tensor
• 3 rd rank Pockels (electrooptic) effect zijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=zijkEk
• Relates Electric field E with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2]V
• Symmetrized indexes due to intrinsic symmetry:
• zijk = zjik
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Information about the selected tensor
• 3 rd rank Thermoelectro-optical effect tensor r(T)ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=r(T)ijkEkΔT
• Relates Electric field E and Temperature variation ΔT with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2]V
• Symmetrized indexes due to intrinsic symmetry:
• r(T)ijk = r(T)jik
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Information about the selected tensor
• 3 rd rank Magneto-optical tensor (Faraday effect) Fijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=FijkHk
• Relates Magnetic field H with the antisymmetric part of the Dielectric impermeability tensor variation Δβij.
• Pure imaginary in non-dissipative media.
• Intrinsic symmetry symbol: e{V2}V
• Symmetrized indexes due to intrinsic symmetry:
• Fijk = -Fjik
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Information about the selected tensor
• 3 rd rank Thermomagneto-optical effect tensor f(T)ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=f(T)ijkHkΔT
• Relates Magnetic field H and Temperature variation ΔT with the antisymmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: e{V2}V
• Symmetrized indexes due to intrinsic symmetry:
• f(T)ijk = -f(T)jik
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Information about the selected tensor
• 3 rd rank Electrogyration effect tensor γijk γijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δgij=γijkEk
• Relates Electric field E with Optical activity tensor variation Δgij
• Intrinsic symmetry symbol: e[V2]V
• Symmetrized indexes due to intrinsic symmetry:
• γijk = γjik
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Information about the selected tensor
• 4 th rank Birefringence in Cubic Crystals γijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij= γijlmklkm
• Relates light wave vector and light wave vector with dielectric impermeability tensor variation (symmetric part).
• Real in non-dissipative media.
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• γijkl = γjikl
• γijkl = γijlk
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Information about the selected tensor
• 4 th rank Elasto-optical tensor pijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=pijklεkl
• Relates Strain tensor εij with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• πijkl = πjikl
• πijkl = πijlk
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Information about the selected tensor
• 4 th rank Kerr effect tensor Rijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=RijklEkEl
• Relates Electric field E and Electric field E with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• Rijkl = Rjikl
• Rijkl = Rijlk
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Information about the selected tensor
• 4 th rank Piezo-optical tensor πijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=πijklσkl
• Relates Stress tensor σij with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• πijkl = πjikl
• πijkl = πijlk
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Information about the selected tensor
• 4 th rank Second-order magneto-optical (Cotton-Mouton effect) tensor Cijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=CijklHkHl
• Relates Magnetic field H and Magnetic field H with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• Cijkl = Cjikl
• Cijkl = Cijlk
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Information about the selected tensor
• 4 th rank Thermopiezo-optical effect tensor π(T)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=π(T)ijklσklΔT
• Relates Stress tensor σij and Temperature variation ΔT with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• π(T)ijkl = π(T)jikl
• π(T)ijkl = π(T)ijlk
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Information about the selected tensor
• 4 th rank Magnetoelectro-optical effect tensor mijkl
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=mijklHkEl
• Relates Magnetic field H and Electric field E with the antisymmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: e{V2}V2
• Symmetrized indexes due to intrinsic symmetry:
• mijkl = -mjikl
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Information about the selected tensor
• 4 th rank Quadratic electrogyration effect tensor βijkl βijkl
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δgij=βijklEkEl
• Relates Electric field E and Electric field E with Optical activity tensor variation Δgij
• Intrinsic symmetry symbol: e[V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• βijkl = βjikl
• βijkl = βijlk
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Information about the selected tensor
• 4 th rank Piezogyration tensor Cijkl
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δgij=Cijklσkl
• Relates Stress tensor σij with Optical activity tensor variation Δgij
• Intrinsic symmetry symbol: e[V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• Cijkl = Cjikl
• Cijkl = Cijlk
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Information about the selected tensor
• 5 th rank Piezoelectro-optical effect tensor zijklm
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=zijklmσklEm
• Relates Stress tensor σij and Electric field E with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2][V2]V
• Symmetrized indexes due to intrinsic symmetry:
• zijklm = zjiklm
• zijklm = zijlkm
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Information about the selected tensor
• 5 th rank Piezomagneto-optical effect tensor ωijklm
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=ωijklmHkσlm
• Relates Magnetic field H and Stress tensor σij with the antisymmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: e{V2}V[V2]
• Symmetrized indexes due to intrinsic symmetry:
• ωijklm = -ωjiklm
• ωijklm = ωijkml
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Information about the selected tensor
• 5 th rank Gradient piezogyration tensor βijklm
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δgij=βijklm∇mσkl
• Relates Stress tensor gradient ∇kσij with Optical activity tensor variation Δgij
• Intrinsic symmetry symbol: e[V2][V2]V
• Symmetrized indexes due to intrinsic symmetry:
• βijklm = βjiklm
• βijklm = βijlkm
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Information about the selected tensor
• 6 th rank Second-order piezo-optical tensor Πijklmn
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δβij=Πijklmnσklσmn
• Relates Stress tensor σij and Stress tensor σij with the symmetric part of the Dielectric impermeability tensor variation Δβij
• Intrinsic symmetry symbol: [V2][[V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• Πijklmn = Πjiklmn
• Πijklmn = Πijlkmn
• Πijklmn = Πijklnm
• Πijklmn = Πijmnkl
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Information about the selected tensor
• 6 th rank Quadratic piezogyration tensor Cijklmn
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Δgij=Cijklmnσklσmn
• Relates Stress tensor σij and Stress tensor σij with Optical activity tensor variation Δgij
• Intrinsic symmetry symbol: e[V2][[V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• Cijklmn = Cjiklmn
• Cijklmn = Cijlkmn
• Cijklmn = Cijklnm
• Cijklmn = Cijmnkl
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Information about the selected tensor
• 3 rd rank General second-order susceptibility (non dissipative media and no dispersion) χ(ω3;ω2,ω1)ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)
• Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Polarization of frequency ω3.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry
• Intrinsic symmetry symbol: [V3]
• Symmetrized indexes due to intrinsic symmetry:
• χ(ω3;ω2,ω1)ijk = χ(ω3;ω2,ω1)kji
• χ(ω3;ω2,ω1)ijk = χ(ω3;ω2,ω1)jik
• χ(ω3;ω2,ω1)ijk = χ(ω3;ω2,ω1)ikj
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Information about the selected tensor
• 3 rd rank Optical rectification (non-dissipative media and no dispersion) χ(0;ω,-ω)ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V3]
• Symmetrized indexes due to intrinsic symmetry:
• χ(0;ω,-ω)ijk = χ(0;ω,-ω)kji
• χ(0;ω,-ω)ijk = χ(0;ω,-ω)jik
• χ(0;ω,-ω)ijk = χ(0;ω,-ω)ikj
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Information about the selected tensor
• 3 rd rank Second-harmonic generation (non-dissipative media and no dispersion) χ(2ω;ω,ω)ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V3]
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)kji
• χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)jik
• χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)ikj
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Information about the selected tensor
• 3 rd rank General optical rectification (dissipative media) χ(0;ω,-ω)ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(0)=χijk(0;ω,-ω)Ej(ω)Ek(-ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 0.
• Tensor of complex coefficients. Real in non-dissipative media.
• Intrinsic symmetry symbol: V[V2]
• Symmetrized indexes due to intrinsic symmetry:
• χ(0;ω,-ω)ijk = χ(0;ω,-ω)ikj
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Information about the selected tensor
• 3 rd rank General second-harmonic generation χ(2ω;ω,ω)ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(2ω)=χijk(2ω;ω,ω)Ej(ω)Ek(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of complex coefficients (imaginary part null). Real in non-dissipative media.
• Intrinsic symmetry symbol: V[V2]
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;ω,ω)ijk = χ(2ω;ω,ω)ikj
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Information about the selected tensor
• 3 rd rank General second-order susceptibility χ(ω3;ω2,ω1)ijk
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(ω3)=χijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)
• Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Polarization of frequency ω3.
• Tensor of complex coefficients.Real in non-dissipative media
• Intrinsic symmetry symbol: V3
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Information about the selected tensor
• 3 rd rank General second-order susceptibility by magnetic dipole (non-dissipative media and no dispersion) χm(ω3;ω2,ω1)ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Mi(ω3)=χmijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)
• Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Magnetization of frequency ω3.
• Tensor of pure imaginary coefficients. Real part null. Kleinman symmetry.
• Intrinsic symmetry symbol: e[V3]
• Symmetrized indexes due to intrinsic symmetry:
• χm(ω3;ω2,ω1)ijk = χm(ω3;ω2,ω1)kji
• χm(ω3;ω2,ω1)ijk = χm(ω3;ω2,ω1)jik
• χm(ω3;ω2,ω1)ijk = χm(ω3;ω2,ω1)ikj
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Information about the selected tensor
• 3 rd rank General second-order susceptibility by magnetic dipole (non-dissipative media) χm(ω3;ω2,ω1)ijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Mi(ω3)=χmijk(ω3;ω2,ω1)Ej(ω2)Ek(ω1)
• Relates Electric field of frequency ω1 and Electric field of frequency ω2 with Magnetization of frequency ω3.
• Tensor of pure imaginary coefficients. Real part null.
• Intrinsic symmetry symbol: eV3
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Information about the selected tensor
• 4 th rank Degenerate four-wave mixing χ(ω;-ω,ω,ω)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(ω=χijkl(ω;-ω,ω,ω)Ej(-ω)Ek(ω)El(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency ω.
• Tensor of complex coefficients.
• Intrinsic symmetry symbol: [[V2][V2]]
• Symmetrized indexes due to intrinsic symmetry:
• χ(ω;-ω,ω,ω)ijkl = χ(ω;-ω,ω,ω)jikl
• χ(ω;-ω,ω,ω)ijkl = χ(ω;-ω,ω,ω)ijlk
• χ(ω;-ω,ω,ω)ijkl = χ(ω;-ω,ω,ω)klij
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Information about the selected tensor
• 4 th rank Electric-field induced second-harmonic generation (non-dissipative media and no dispersion) χ(2ω;0,ω,ω)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(2ω=χijkl(2ω;0,ω,ω)Ej(0)Ek(ω)El(ω)
• Relates Electric field of frequency 0 and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V4]
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ljki
• χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)kjil
• χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)jikl
• χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ilkj
• χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ikjl
• χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ijlk
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Information about the selected tensor
• 4 th rank General third-order susceptibility (non-dissipative media and no dispersion) χ(ω4;ω3,ω2,ω1)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1)
• Relates Electric field of frequency ω1 and Electric field of frequency ω2 and Electric field of frequency ω3 with Polarization of frequency ω4.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V4]
• Symmetrized indexes due to intrinsic symmetry:
• χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)ljki
• χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)kjil
• χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)jikl
• χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)ilkj
• χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)ikjl
• χ(ω4;ω3,ω2,ω1)ijkl = χ(ω4;ω3,ω2,ω1)ijlk
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Information about the selected tensor
• 4 th rank Third-harmonic generation (non-dissipative media and no dispersion) χ(3ω;ω,ω,ω)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω.
• Tensor of real coefficients (imaginary part null). Kleinman symmetry.
• Intrinsic symmetry symbol: [V4]
• Symmetrized indexes due to intrinsic symmetry:
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ljki
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)kjil
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)jikl
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ilkj
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ikjl
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ijlk
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Information about the selected tensor
• 4 th rank Four-wave mixing χ(ω1;-ω2,ω1,ω2)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(ω1)=χijkl(ω1;-ω2,ω1,ω2)Ej(-ω2)Ek(ω1)El(ω2)
• Relates Electric field of frequency ω1 and Electric field of frequency ω2 and Electric field of frequency ω2 with Polarization of frequency ω1.
• Tensor of complex coefficients.
• Intrinsic symmetry symbol: [V2V2]
• Symmetrized indexes due to intrinsic symmetry:
• χ(ω1;-ω2,ω1,ω2)ijkl = χ(ω1;-ω2,ω1,ω2)klij
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Information about the selected tensor
• 4 th rank General third-harmonic generation (dissipative media) χ(3ω;ω,ω,ω)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(3ω)=χijkl(3ω;ω,ω,ω)Ej(ω)Ek(ω)El(ω)
• Relates Electric field of frequency ω and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 3ω.
• Tensor of complex coefficients. real for non-dissipative media.
• Intrinsic symmetry symbol: V[V3]
• Symmetrized indexes due to intrinsic symmetry:
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ilkj
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ikjl
• χ(3ω;ω,ω,ω)ijkl = χ(3ω;ω,ω,ω)ijlk
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Information about the selected tensor
• 4 th rank General third-order susceptibility χ(ω4;ω3,ω2,ω1)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(ω4)=χijkl(ω4;ω3,ω2,ω1)Ej(ω3)Ek(ω2)El(ω1)
• Relates Electric field of frequency ω1 and Electric field of frequency ω2 and Electric field of frequency ω3 with Polarization of frequency ω4.
• Tensor of complex coefficients. Real for non-dissipative media.
• Intrinsic symmetry symbol: V4
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Information about the selected tensor
• 4 th rank Electric-field induced second-harmonic generation (non-dissipative media) χ(2ω;0,ω,ω)ijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Pi(2ω=χijkl(2ω;0,ω,ω)Ej(0)Ek(ω)El(ω)
• Relates Electric field of frequency 0 and Electric field of frequency ω and Electric field of frequency ω with Polarization of frequency 2ω.
• Tensor of real coefficients (imaginary part null).
• Intrinsic symmetry symbol: V2[V2]
• Symmetrized indexes due to intrinsic symmetry:
• χ(2ω;0,ω,ω)ijkl = χ(2ω;0,ω,ω)ijlk
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Information about the selected tensor
• 2 nd rank Diffusion tensor Dij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ji=Dij∇jC
• Relates Concentration gradient ∇C with Diffusive flux J
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• Dij = Dji
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Information about the selected tensor
• 2 nd rank Dufour effect (reversal thermodiffusion) tensor βij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: qi=βij∇jC
• Relates Concentration gradient ∇C with Heat flux q
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• βij = βji
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Information about the selected tensor
• 2 nd rank Electric conductivity tensor σij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ji=σijEj
• Relates Electric field E with Electric current density J
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• σij = σji
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Information about the selected tensor
• 2 nd rank Electric resistivity tensor ρij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ei=ρijJj
• Relates Electric current density J with Electric field E
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• ρij = ρji
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Information about the selected tensor
• 2 nd rank Electrodiffusion tensor γij (direct effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ji=γijEjT
• Relates Electric field E with Diffusive flux J
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• γij = γji
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Information about the selected tensor
• 2 nd rank Electrodiffusion tensor γTij (inverse effect)
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ji=γTij∇jC
• Relates Concentration gradient ∇C with Electric current density J
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• γTij = γTji
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Information about the selected tensor
• 2 nd rank Soret effect (thermodiffusion) tensor βij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ji=βij∇jT
• Relates Temperature gradient ∇T with Diffusive flux J
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• βij = βji
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Information about the selected tensor
• 2 nd rank Thermal conductivity tensor κij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: qi=κij∇Tj
• Relates Temperature gradient ∇T with Heat flux q
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• κij = κji
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Information about the selected tensor
• 2 nd rank Thermal diffusivity tensor αij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ∂T/∂t=αij∇T∇T
• Relates Temperature gradient ∇T and Temperature gradient ∇T with Time derivative of the tempreature ∂T/∂t
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• αij = αji
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Information about the selected tensor
• 2 nd rank Thermal resistivity tensor rij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ∇Ti=rijqj
• Relates Heat flux q with Temperature gradient ∇T
• Intrinsic symmetry symbol: [V2]
• Symmetrized indexes due to intrinsic symmetry:
• rij = rji
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Information about the selected tensor
• 2 nd rank Peltier effect tensor πij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: qi=πijJj
• Relates Electric current density J with Heat flux q
• Intrinsic symmetry symbol: V2
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Information about the selected tensor
• 2 nd rank Thermoelectric power (Seebeck effect) tensor βij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ei=βij∇Tj
• Relates Temperature gradient ∇T with Electric field E
• Intrinsic symmetry symbol: V2
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Information about the selected tensor
• 2 nd rank Thomson heat tensor τij
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: ∂q/∂t=τij∇TiJj
• Relates Temperature gradient ∇T and Electric current density J with Heat production rate ∂q/∂t
• Intrinsic symmetry symbol: V2
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Information about the selected tensor
• 3 rd rank Hall effect (magnetorresistance) tensor Rijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Ei=RijkJjHk
• Relates Electric current density J and Magnetic field H with Electric field E
• Intrinsic symmetry symbol: e{V2}V
• Symmetrized indexes due to intrinsic symmetry:
• Rijk = -Rjik
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Information about the selected tensor
• 3 rd rank Righi-Leduc, Maggi-Righi-Leduc and magnetothermal effects tensor Qijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: qi=Qijk∇TjHk
• Relates Temperature gradient ∇T and Magnetic field H with Heat flux q
• Intrinsic symmetry symbol: e{V2}V
• Symmetrized indexes due to intrinsic symmetry:
• Qijk = -Qjik
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Information about the selected tensor
• 3 rd rank Ettinghausen effect tensor Mijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: qi=MijkJjHk
• Relates Electric current density J and Magnetic field H with Heat flux q
• Intrinsic symmetry symbol: eV3
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Information about the selected tensor
• 3 rd rank Nernst effect tensor Nijk
• Axial tensor invariant under time-reversal symmetry operation
• Defining equation: Ei=Nijk∇TjHk
• Relates Temperature gradient ∇T and Magnetic field H with Electric field E
• Intrinsic symmetry symbol: eV3
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Information about the selected tensor
• 4 th rank Magnetic resistance tensor Tijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ei=TijklJjHkHl
• Relates Electric current density J and Magnetic field H and Magnetic field H with Electric field E
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• Tijkl = Tjikl
• Tijkl = Tijlk
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Information about the selected tensor
• 4 th rank Piezoresistivity (Strain Gauge effect) tensor πijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Δρij=πijklσkl
• Relates Stress tensor σij with Electric resistivity tensor variation Δρij
• Intrinsic symmetry symbol: [V2][V2]
• Symmetrized indexes due to intrinsic symmetry:
• πijkl = πjikl
• πijkl = πijlk
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Information about the selected tensor
• 4 th rank Magneto Peltier effect Pijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: qi=PijklHkHlJj
• Relates current density and magnetic field and magnetic field with heat flux.
• Intrinsic symmetry symbol: V2[V2]
• Symmetrized indexes due to intrinsic symmetry:
• Pijkl = Pijlk
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Information about the selected tensor
• 4 th rank Magneto Seebeck effect αijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ei=αijklHkHl(ΔT)j
• Relates temperature gradient ΔT and magnetic field and magnetic field with electric field.
• Intrinsic symmetry symbol: V2[V2]
• Symmetrized indexes due to intrinsic symmetry:
• αijkl = αijlk
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Information about the selected tensor
• 4 th rank Magneto-Seebeck effect tensor αijkl
• Polar tensor invariant under time-reversal symmetry operation
• Defining equation: Ei=αijkl∇TjHkHl
• Relates Temperature gradient ∇T and Magnetic field H and Magnetic field H with Electric field E
• Intrinsic symmetry symbol: V2[V2]
• Symmetrized indexes due to intrinsic symmetry:
• αijkl = αijlk
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TENSOR